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The two fundamental kinds of quantification in predicate logic are universal quantification and existential quantification. These concepts are covered in detail in their individual articles; here we discuss features of quantification that apply in both cases. Other kinds of quantification include uniqueness quantification.

1 Need for quantification in natural language

Natural language makes frequent use of quantification:

• Every glass in my recent order was chipped.
• Some of the people standing across the river have white armbands.
• Most of the people I talked to didn't have a clue who the candidates were.
• Everyone in the waiting room had at least one complaint against Dr. Ballyhoo.
• There was somebody in his class that was able to correctly answer every one of the questions I submitted.
• A lot of people are idiots.

Linguists regard even some slang or scatological expressions semantically as quantifiers:

• There were a shitload of problems with that new software we bought.

There exists no simple way of reformulating any one of these expressions as a conjunction or disjunction of sentences, each a simple predicate of an individual such as That wine glass was chipped. These examples also suggest that the construction of quantified expressions in natural language can be syntactically very complicated. Fortunately, for mathematical assertions, the quantification process is syntactically more straightforward.

The study of quantification in natural languages is much more difficult than the corresponding problem for formal languages. This comes in part from the fact that the grammatical structure of natural language sentences may conceal the logical structure. Moreover, mathematical conventions strictly specify the range of validity for formal language quantifiers; for natural language, specifying the range of validity requires dealing with non-trivial semantic problems.

2 Need for quantifiers in mathematical assertions

We will begin by discussing quantification in informal mathematical discourse. Consider the following statement

1·2 = 1 + 1, and 2·2 = 2 + 2, and 3 . 2 = 3 + 3, ...., and n · 2 = n + n, etc.

This has the appearance of an infinite conjunction of propositions. From the point of view of formal languages this is immediately a problem, since we expect syntaxThe first meaning of the term syntax originating from the Greek words (sun, meaning ‘together’) and (taxis, meaning sequence/order), can be described as the study of the rules, or "patterned relations" that govern the way the words in a sentence come toge rules to generate finiteIn mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2,. n with n isin N . It is a theorem that a set is finite if and only if there exists no bijection between the set and any of its prop objects. Putting aside this objection, also note that in this example we were lucky in that there is a procedureA procedure is a series of activities, tasks, steps, decisions, calculations and other processes, that when undertaken in the sequence laid down produces the described result, product or outcome. Following a procedure should produce repeatable results for to generate all the conjuncts. However, if we wanted to assert something about every irrational numberIn mathematics, an irrational number is any real number that is not a rational number, i. one that cannot be written as a fraction a ''b with a and b integers, and b not zero. It can readily be shown that the irrational numbers are precisely those numbers, we would have no way enumerating all the conjuncts since irrationals cannot be enumerated. A succinct formulation which avoids these problems uses universal quantification:

For any natural numberNatural number can mean either a positive integer ( 1, 2, 3, 4,. or a non-negative integer ( 0, 1, 2, 3, 4,. Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), or they can be used for ordering ("this n, n·2 = n + n.

A similar analysis applies to the disjunction,

1 is primeIn mathematics, a prime number or prime for short, is a natural number whose only distinct positive divisors are 1 and itself; otherwise it is called a composite number . Hence a prime number has exactly two divisors. The number 1 is neither prime nor com, or 2 is prime, or 3 is prime, etc.

which can be rephrased using existential quantification:

For some natural numberNatural number can mean either a positive integer ( 1, 2, 3, 4,. or a non-negative integer ( 0, 1, 2, 3, 4,. Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), or they can be used for ordering ("this n, n is prime.

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Topics: Quantification Language Natural Interpretation Predicate...

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